naginterfaces.library.specfun.jacellip_​real

naginterfaces.library.specfun.jacellip_real(u, m)

jacellip_real evaluates the Jacobian elliptic functions sn, cn and dn.

For full information please refer to the NAG Library document for s21ca

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s21caf.html

Parameters

ufloat

The argument $u$.

mfloat

The argument $m$.

Returns

snfloat

The value of $sn\left(u\right)$.

cnfloat

The value of $cn\left(u\right)$.

dnfloat

The value of $dn\left(u\right)$.

Raises

NagValueError

(errno $1$)

On entry, $\left|u\right|$ is too large: $\left|u\right|=⟨value⟩$ it must be less than $⟨value⟩$.

(errno $2$)

On entry, $\left|m\right|$ is too large when used in conjunction with the supplied argument $u$: $\left|m\right|=⟨value⟩$ it must be less than $⟨value⟩$.

Notes

jacellip_real evaluates the Jacobian elliptic functions of argument $u$ and argument $m$,

$$\begin{array}{c}\begin{array}{ccc}sn\left(u|m\right)& =& \sin \left(\varphi \right)\text{,}\\ cn\left(u|m\right)& =& \cos \left(\varphi \right)\text{,}\\ dn\left(u|m\right)& =& \sqrt{1-m{\sin }^2\left(\varphi \right)}\text{,}\end{array}\end{array}$$

where $\varphi$, called the amplitude of $u$, is defined by the integral

$$u={\int }_0^{\varphi }\frac{d\theta }{\sqrt{1-m{\sin }^2\left(\theta \right)}}\text{.}$$

The elliptic functions are sometimes written simply as $sn\left(u\right)$, $cn\left(u\right)$ and $dn\left(u\right)$, avoiding explicit reference to the argument $m$.

Another nine elliptic functions may be computed via the formulae

$$\begin{array}{c}\begin{array}{ccc}cd\left(u\right)& =& cn\left(u\right)/dn\left(u\right)\\ sd\left(u\right)& =& sn\left(u\right)/dn\left(u\right)\\ nd\left(u\right)& =& 1/dn\left(u\right)\\ dc\left(u\right)& =& dn\left(u\right)/cn\left(u\right)\\ nc\left(u\right)& =& 1/cn\left(u\right)\\ sc\left(u\right)& =& sn\left(u\right)/cn\left(u\right)\\ ns\left(u\right)& =& 1/sn\left(u\right)\\ ds\left(u\right)& =& dn\left(u\right)/sn\left(u\right)\\ cs\left(u\right)& =& cn\left(u\right)/sn\left(u\right)\end{array}\end{array}$$

(see Abramowitz and Stegun (1972)).

jacellip_real is based on a procedure given by Bulirsch (1960), and uses the process of the arithmetic-geometric mean ($16.9$ in Abramowitz and Stegun (1972)). Constraints are placed on the values of $u$ and $m$ in order to avoid the possibility of machine overflow.

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Bulirsch, R, 1960, Numerical calculation of elliptic integrals and elliptic functions, Numer. Math. (7), 76–90